Steady states of FitzHugh-Nagumo-type systems with sign-changing coefficients
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arXiv
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| Autori principali: | , , |
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| Natura: | Preprint |
| Pubblicazione: |
2025
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| _version_ | 1866908496464707584 |
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| author | Ó, João Marcos do Shamarova, Evelina Silva, Victor V. |
| author_facet | Ó, João Marcos do Shamarova, Evelina Silva, Victor V. |
| contents | We establish existence and multiplicity results for steady-state solutions of spatially heterogeneous FitzHugh-Nagumo-type systems, extending the existing theory from constant to variable coefficients that may change sign. Specifically, we study the system Specifically, we study the system \begin{align*}
-Δu + a(x)v &= f(x,u) && \text{in } \mathbb{R}^N, \\ -Δv + b(x)v &= c(x)u && \text{in } \mathbb{R}^N. \end{align*} where $N \geqslant 3$, the coefficients $a,b,c : \mathbb{R}^N \to \mathbb{R}$ are $L^\infty_{\mathrm{loc}}$-functions bounded from below, and $f:\mathbb{R}^N \times \mathbb{R} \to \mathbb{R}$ is a Carathéodory function with subcritical growth. For assumptions permitting sign changes and non-coercivity of the coefficients, we prove the existence of a mountain pass solution. In the case where $a,b,c$ do not change sign, still allowing non-coercive behavior, we additionally establish the existence of componentwise positive and negative solutions. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2508_14854 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Steady states of FitzHugh-Nagumo-type systems with sign-changing coefficients Ó, João Marcos do Shamarova, Evelina Silva, Victor V. Analysis of PDEs 35J60, 35B09, 35J30 We establish existence and multiplicity results for steady-state solutions of spatially heterogeneous FitzHugh-Nagumo-type systems, extending the existing theory from constant to variable coefficients that may change sign. Specifically, we study the system Specifically, we study the system \begin{align*} -Δu + a(x)v &= f(x,u) && \text{in } \mathbb{R}^N, \\ -Δv + b(x)v &= c(x)u && \text{in } \mathbb{R}^N. \end{align*} where $N \geqslant 3$, the coefficients $a,b,c : \mathbb{R}^N \to \mathbb{R}$ are $L^\infty_{\mathrm{loc}}$-functions bounded from below, and $f:\mathbb{R}^N \times \mathbb{R} \to \mathbb{R}$ is a Carathéodory function with subcritical growth. For assumptions permitting sign changes and non-coercivity of the coefficients, we prove the existence of a mountain pass solution. In the case where $a,b,c$ do not change sign, still allowing non-coercive behavior, we additionally establish the existence of componentwise positive and negative solutions. |
| title | Steady states of FitzHugh-Nagumo-type systems with sign-changing coefficients |
| topic | Analysis of PDEs 35J60, 35B09, 35J30 |
| url | https://arxiv.org/abs/2508.14854 |